Hybrid Fault diagnosis capability analysis of Hypercubes under the PMC model and MM model

Hybrid Fault diagnosis capability analysis of Hypercubes under the PMC model and MM model Qiang Zhu,Lili Li, Sanyang Liu, Xing Zhang arxiv:1709.05588v1 [cs.dc] 17 Sep 017 Abstract System level diagnosis is an important approach for the fault diagnosis of multiprocessor systems. In system level diagnosis, diagnosability is an important measure of the diagnosis capability of interconnection networks. But as a measure, diagnosability can not reflect the diagnosis capability of multiprocessor systems to link faults which may occur in real circumstances. In this paper, we propose the definition of h-edge tolerable diagnosability to better measure the diagnosis capability of interconnection networks under hybrid fault circumstances. The h-edge tolerable diagnosability of a multiprocessor system G is the maximum number of faulty nodes that the system can guarantee to locate when the number of faulty edges does not exceed h,denoted by t e h(g). The PMC model and MM model are the two most widely studied diagnosis models for the system level diagnosis of multiprocessor systems. The hypercubes are the most well-known interconnection networks. In this paper, the h-edge tolerable diagnosability of n-dimensional hypercube under the PMC model and MM is determined as follows: t e h(q n) = n h, where 1 h < n, n 3. Index Terms hypercubes, PMC model, MM model, diagnosability, fault diagnosis, multiprocessor interconnection networks The authors are with the School of Mathematics and Statistics, Xidian University, Xi an, Shaanxi 710071, China. E-mail: zhuustcer@gmail.com,liusanyang@16.com Supported by National Natural Science Foundation of China (Nos: 616705, 61674108, 61373174)

1 Hybrid Fault diagnosis capability analysis of Hypercubes under the PMC model and MM model I. INTRODUCTION Nowadays, some supercomputers have hundreds of thousands of processors. It s inevitable that some node or link faults may occur in such large systems. System level diagnosis is an approach for the fault diagnosis of multiprocessor systems by means of mutual self-tests of processors in the system. Distinct definitions of test and distinct assumptions on test result lead to distinct diagnosis models. PMC model [1] (introduced by Preparata, Metze and Chien) is the most widely studied model in system level diagnosis. Under the PMC model, a test involves two processors: the tester and the testee. It s assumed that only adjacent nodes can test the status of each other. As to the test results, it is assumed that if the tester is fault-free(resp. faulty), then the test result is reliable(resp. unreliable). Another important model, first proposed by Malek and Maeng [], is called the comparison diagnosis model (MM model). It is well known and widely studied in recent years. In this model, a test t(u, v; w) involves 3 processors u, v, w where w is the common neighbor of u, v. w is called the comparator of u, v, u, v are called the compared vertices. Under the MM model, it is assumed that the test result r(u, v; w) is reliable if and only if the comparator w is fault-free. That is, if w is faultfree, r(u, v; w) = 0 if and only if both u, v are fault-free. If w is faulty, r(u, v; w) may be 0 or 1 irrelevant to the status of u, v. In 199, Sengupta and Dahbura [3] suggested a further modification of the MM model, called the MM model, in which every node must compare every pair of its distinct neighbors. MM model is widely used in the diagnosis capability analysis of interconnection networks [4] [14]. The performance of a large multiprocessor system is much impacted by its underlying topology which can be modeled by a graph called its interconnection network when each processor is regarded as a vertex and each communication link is regarded as an edge. In the past two decades, various interconnection networks have been proposed. To choose an appropriate one, the properties of these interconnection networks have to be explored. Among all the properties of an interconnection network, fault diagnosis capability is very important to its suitability as the underlying topology of a high performance fault tolerant computing system. Diagnosability, conditional diagnosability, g-good neighbor conditional diagnosability and h-extra diagnosability are some parameters for measuring the diagnosis capability of interconnection networks.the diagnosability of a multiprocessor system is the maximum number of faulty nodes that the system can guarantee to locate. That is, suppose the diagnosability of a system G is t, then for all circumstances where the number of faulty nodes in G does not exceed t, all these faulty nodes can be located in one step and there exists a circumstance with t + 1 faulty vertices such that not all faulty nodes can be located in one step. As an important measure of the diagnosis capability, the diagnosability of various interconnection networks have been studied [6], [8], [15] [1]. Diagnosability is a worst-case measure, in real circumstances many interconnection networks exhibits much higher fault diagnosis capability compared with their diagnosability []. To better measure the diagnosis capability of such interconnection networks, some new parameters like conditional diagnosability, g-good neighbor diagnosability and h-extra diagnosability have been proposed and studied. For some large interconnection networks like hypercubes, all the neighbours of any node are faulty at the same time is quite small. So conditional diagnosability has been proposed by Lai et al. in 005 [3] by restricting that any node in the multiprocessor system must have a fault-free neighbor. The conditional diagnosability of various interconnection networks have been investigated [10], [11], [3] [33]. In 01, Peng et al. proposed g-good neighbor conditional diagnosability as a generalization of the conditional diagnosability by assuming that any node in a multiprocessor system much have at least g- good neighbors [34]. The g-good conditional diagnosability of several interconnection networks have been explored. In 015, Zhang et al. [1] proposed g-extra conditional diagnosability by assuming that when some nodes fail there are no components with less than g+1 vertices in the remaining system. The g-extra diagnosability of several well-known interconnection networks such as hypercubes, fold hypercubes, arrangement graph and bubble-sort star graph have been studied [1] [14], [35]. Although these parameters may better measure the diagnosis capability of interconnection networks, they are all only applicable to the circumstance with no link faults in the system. But in real circumstance, both node and link faults may occur when a network is put into use. It is significant to study the diagnosis capability of multiprocessor systems with both node and link faults. In 1998, CL Yang and GM Masson studied the hybrid fault diagnosability of multiprocessor systems with unreliable communication links by introducing bounded number of incorrect test results [36]. In 000, D Wang discussed the strategies for determining the diagnosability of hypercubes with arbitrary missing edges under both the PMC model and BGM model [37]. In 007, S Zheng and S Zhou proved that under the PMC model and the BGM model, the diagnosability of star graphs with missing edges can be determined by its minimum degree [38]. In 01, CF Chiang et al. showed that under the MM model the n-dimensional star graph preserves the strong local diagnosability when the number of faulty links does not exceed n 3 [6].

To better adapt to the real circumstances where both node and link faults may occur, we introduce the h-edge tolerable diagnosability to better measure the diagnosis capability of multiprocessor systems. Then we investigate the h-edge tolerable diagnosability of hypercubes under the PMC model and MM model. The rest of this paper is organized as follows. Section introduces some definitions, notations and terminologies; In section 3, Preliminaries on the PMC and MM model, the concept of h-edge tolerable diagnosability and some basic results about it are presented. In section 4, the h-edge tolerable diagnosability of the hypercube under the PMC model and MM model are explored. Section 4 summarizes the results of this paper and gives some possible future directions. II. NOTATIONS AND TERMINOLOGIES We follow [39] for terminologies and notations not defined here. Let G = (V (G), E(G)) be a simple undirected graph, V (G) and E(G) are used to denote its vertex set and edge set respectively. We use the unordered pair (u, v) for an edge e = {u, v}. If (u, v) E(G), then u and v are adjacent and u, v are incident to the edge (u, v). We use E G (u) (resp. N G (u)) to denote the set of edges incident (resp. vertices adjacent) to u in G. The number of vertices in N G (u) is called the degree of u in G, denoted by deg G (u). The minimum degree of G is the minimum over all degrees of the vertices in G, denoted by δ(g). If for any vertex u G, deg G (u) = k, then the graph is said to be k-regular. When G is clear from the context, we use E(u), N(u) and d(u) to respectively replace E G (u) and N G (u) and deg G (u) for simplification. For any nonempty vertex set S of G, we use N G (S) to denote the set of vertices in V (G) S which are adjacent to at least one vertex in S, that is, N G (S) = {u V (G) S v S such that (u, v) E(G)}. Let C G (S) = N G (S) S. For two graphs H and G, we say H is a subgraph of G if V (H) V (G) and E(H) E(G), denoted by H G. The vertex boundary number of H is the number of vertices in N G (V (H)), denoted by b v (H, G). By definition, it s obvious that subgraphs with the same vertex set have the same boundary number. The minimum m-boundary number of G is defined as the minimum boundary number of all its subgraphs with order m, denoted by δ v (m; G). That is, δ v (m; G) = {b v (H, G) H G and V (H) = m}. Given a graph G = (V, E), the symmetric difference of any two vertex subsets A and B is the set of elements in exactly one of A, B, denoted by A B. That is, A B = (A B) (A B). The complement of A in G is V (G) A, denoted by A. III. PRELIMINARIES AND BASICS OF THE h-edge TOLERABLE DIAGNOSABILITY A. Preliminaries on The PMC model In 1967, Preparata, Metze and Chien [1] proposed a diagnosis model called the PMC model which is the most famous and most widely studied model in the system level diagnosis of multiprocessor systems. Under this model, it is assumed that only node faults can occur and all node faults are permanent. Under this model, a test involves two adjacent processors a tester and a testee. It is assumed that the status of the testee can always be detected by a fault-free tester. A test (u, v) where u is the tester and v is the testee can be represented by an ordered pair (u, v). The test result of (u, v) is denoted by r(u, v). r(u, v) = 0 if u evaluates v as fault-free and r(u, v) = 1 if u evaluates v as faulty. Under the PMC model, it is assumed that test result r(u, v) is reliable if and only if the tester u is fault-free. That is, if u is fault-free, then r(u, v) = 0 means that v is fault-free and r(u, v) = 1 means that v is faulty; If the tester u is faulty, then the test result r(u, v) may be 0 or 1 whether v is faulty or fault-free. B. Preliminaries on The MM model Proposed by Malek and Maeng [], the comparison diagnosis model (MM model) is one of the most popular diagnosis models. In the MM model, A test (u, v) w is done by sending the same task from a node w(the comparator) to a pair of its distinct neighbors: u and v, then comparing their results. The test result of (u, v) w (r((u, v) w )) is 0 (resp. 1) if w evaluates that the returning results of u and v are consistent (resp. not consistent). Under the MM model, it is assumed that if the comparator w is fault-free, r((u, v) w ) = 0 if u and v are both fault-free and r((u, v) w ) = 1 if at least one of u, v is faulty. If the comparator w is faulty, then the test result may be 0 or 1 irrelevant to the status of u and v. C. Basics about the PMC model and MM model Given a multiprocessor system, the set of all test results is called a syndrome of the multiprocessor system. Under the PMC model, a fault set F is said to be consistent with a syndrome σ if σ can be aroused by the circumstance that all nodes in F are faulty and all nodes not in F are faultfree. Since the test result of a faulty tester(resp.comparator ) is unreliable under the PMC model(resp. MM model), a fault set F can be consistent with many syndromes, the set of all syndromes consistent with F f is denoted by σ(f ). Two faulty sets F 1, F are distinguishable if and only if σ(f 1 ) σ(f ) =. Otherwise, they are indistinguishable. Since the test result of a faulty tester(resp. comparator) under the PMC model(resp. MM model) is unreliable, V (G) is consistent with any syndrome of G under both the PMC model and MM model. Thus to locate faulty vertices, people often suppose there exists an upper bounder for the number of faulty vertices. A multiprocessor system G is t-diagnosable if all the faulty vertices can be guaranteed to be located provided that the number of faulty vertices does not exceed t. The diagnosability of G is the maximum integer t such that G is t-diagnosable. The diagnosability of a multiprocessor system can measure its fault diagnosis capability. In [40] A. Dahbura et. al. characterize a pair of distinguishable faulty sets under the PMC model. Lemma 3.1: For a simple undirected graph G = (V, E) and any two different sets F 1, F V, F 1 and F are distinguishable under the PMC model if and only if there exists an edge between V F 1 F and F 1 F.

3 Under the MM model, Sengupta et al. [3] provide a necessary and sufficient condition to check whether a pair of faulty sets is distinguishable. Lemma 3.: [3] For a simple undirected graph G = (V, E) and any two different sets Fv 1, Fv V, Fv 1 and Fv are distinguishable under the MM model if and only if at least one of the following conditions is satisfied: (1) There exist three vertices u (Fv 1 Fv ) and v, w Fv 1 Fv such that uw, vw E(G). () There exist three vertices u, v (Fv 1 Fv ) and w Fv 1 Fv such that uw, vw E(G). (3) There exist three vertices u, v (Fv Fv 1 ) and w Fv 1 Fv such that uw, vw E(G). By Lemma 3.1 and Lemma 3., we have the following corollary: Corollary 3.3: For a simple undirected graph G = (V, E) and any two different sets Fv 1, Fv V, Fv 1 and Fv are distinguishable under the PMC model if they are distinguishable under the MM model D. The h-edge tolerable diagnosability To better adapt to the real circumstances that link faults may happen, we introduce the h-edge tolerable diagnosability of multiprocessor systems to better measure their diagnosis capability. Given a multiprocessor system G(V, E), we want to evaluate the diagnosis capability of the system when some edges are faulty. To do this, suppose F e is the set of faulty edges in the system.then given F e as the set of faulty edges, under the MM model two faulty vertex sets Fv 1 and Fv are distinguishable if and only if they are distinguishable in G F e. According to Lemma 3.1, the following corollary can be obtained: Corollary 3.4: Given a multiprocessor system G(V, E), let F e be the set of faulty edges in G. Then two faulty vertex sets Fv 1 and Fv are indistinguishable in G F e under the PMC model if and only if in the graph G F e any vertex in V Fv 1 Fv has no neighbor in Fv 1 Fv. According to Lemma 3., the following corollary can be obtained: Corollary 3.5: Given a multiprocessor system G(V, E), let F e be the set of faulty edges in G. Then two faulty vertex sets Fv 1 and Fv are indistinguishable in G F e under the MM model if and only if in the graph G F e any vertex in V Fv 1 Fv has at most one neighbor in Fv 1 Fv or Fv Fv 1 and if it has a neighbor in Fv 1 Fv, then it doesn t has any neighbor in V Fv 1 Fv. Remember that the diagnosability of a multiprocessor system G under the MM model is the maximum integer t such that G is t-diagnosable. To better measure the diagnosis capability of multiprocessor systems with link faults, we propose the definition of h-edge tolerable diagnosability as a generalization of diagnosability under the MM model. Definition 3.6: Given a diagnosis model and a multiprocessor system G, G is h-edge tolerable t-diagnosable under the diagnosis model if any pair of faulty vertex sets Fv 1 and Fv are distinguishable in G F e provided that Fv 1, Fv t and F e h where F e E(G) is any edge subset of G with not more than h edges; The h-edge tolerable diagnosability of G, denoted as t e h (G), is the maximum integer t such that G is h-edge tolerable t-diagnosable. The traditional diagnosability of a multiprocessor system G under the MM model can be viewed as the 0-edge tolerable diagnosability of G. Thus h-edge tolerable diagnosability is a generalization of the traditional diagnosability and can better measure the diagnosis capability of interconnection networks. IV. THE h-edge TOLERABLE DIAGNOSABILITY OF HYPERCUBES UNDER THE PMC AND MM MODEL A. Preliminaries on the hypercubes Hypercubes are the most famous and most widely studied interconnection networks. An n-dimensional hypercube Q n can be modeled as a graph where each vertex is labelled with an n-bit binary string. Any pair of distinct vertices in Q n are adjacent if and only if their labels differ in exactly one bit position. In other words, u = u 1 u u n and v = v 1 v v n are adjacent if and only if there exists a positive integer i {1,,, n} such that u i v i and u j = v j, for each j {1,,, n}/{i}. In this case, we call v an i-th neighbor of u, denoted by u i. Similarly u can be denoted as v i. The edge (u, u i ) is called the i-th incident edge of u. Clearly, Q n is a n-regular graph consisting of n vertices and n n 1 edges, the minimum length of all cycles of Q n equals 4 for n. The properties of Q n have been extensively studied. In [41] Zhu et al. showed that any two different vertices in Q n can either have two common neighbors or none. Lemma 4.1: [41] Any two distinct vertices in V (Q n ) have exactly two common neighbours for n 3 if they have any. The results of the m-minimum boundary number of Q n is useful in this paper. Lemma 4.: [4] δ v (m; Q n ) = { m + (n 1 )m + 1, 1 m n + 1 m + (n 3 )m n +. n + m n By Lemma 4., the following corollary can be obtained. Corollary 4.3: For any two integers m 1, m with 1 m 1 m n, δ v (m ; Q n ) δ v (m 1 ; Q n ) 1. B. Main Results In this section, we will study the h-edge tolerable diagnosability of Q n under both the PMC model and the MM model. Lemma 4.4: For an n-dimensional hypercube Q n with n 3, t e h (Q n) n h where 1 h n under both the PMC model and the MM model. Proof : To prove this Lemma, we only need to construct two distinct faulty vertex sets Fv 1, Fv and a faulty edge set F e satisfying F e h and Fv 1, Fv n h + 1 and Fv 1, Fv are indistinguishable in Q n F e. Let u = 0 n, F e = {(u, u 1 ), (u, u ) (u, u h )}, Fv 1 = {u (h+1), u (h+),, u n }, Fv = {u, u (h+1), u (h+),, u n }. Since in Q n F e there is no edge between V Fv 1 F v and Fv 1 Fv, Fv 1 and Fv

4 are indistinguishable in Q n F e under both the PMC model and the MM model according to Corollary 3.4 and Corollary 3.5. Thus t e h (Q n) < max{ Fv 1, Fv 1 } = n h + 1 under both the PMC model and the MM model. The lemma holds. Next, we show that the above upper bound of t e h (Q n) can be reached under both models. Lemma 4.5: Under both the PMC model and the MM model. t e h (Q n) n h, where 1 h n, n 5. Proof : Suppose, by contradiction, that t e h (Q n) < n h under both models. By Definition 3.6 and Corollary 3.3, there exists F e, E(Q n ) and Fv 1, Fv V (Q n ) with F e h, Fv 1, Fv n h such that Fv 1 Fv are indistinguishable in Q n F e under the MM model. By Theorem 3., in the graph Q n F e any vertex in V Fv 1 Fv has at most one neighbor in Fv 1 Fv or Fv Fv 1 and if it has a neighbor in V Fv 1 Fv then it doesn t has any neighbor in V Fv 1 Fv. So the following proof is divided into two cases: 1) N V F 1 v Fv (F v 1 Fv ) =. ) N V F 1 v Fv (F v 1 Fv ) Case 1). N V F 1 v Fv (F v 1 Fv ) = In this case, N Qn F e (Fv 1 Fv ) Fv 1 Fv. So Fv 1 + Fv Fv 1 Fv = Fv 1 Fv + Fv 1 Fv N Qn F e (Fv 1 Fv ) + Fv 1 Fv. For simplicity, let m = Fv 1 Fv. Subcase 1.1) m = 1. Without loss of generality, suppose Fv 1 Fv, then Fv 1 Fv = Fv. Let {u} = Fv 1 Fv Since N V F 1 v Fv (F v 1 Fv ) =, all the end-vertices of u in Q n F e are in Fv. So Fv 1 n h + 1, a contradiction. Subcase 1.) m =. It s obvious that N Qn F e (Fv 1 Fv ) n h. So n h Fv 1 + Fv Fv 1 Fv N Qn F e (Fv 1 Fv ) + Fv 1 Fv n h + = n h, a contradiction. Subcase 1.) m 3. Since Fv 1, Fv n h, m n h and 1 h n. By Corollary 4.3, it is easy to see that N Qn (Fv 1 Fv ) δ v (3; Q n ) 1 = 3n 6. Therefore, n h Fv 1 Fv N Qn F e (Fv 1 Fv ) + Fv 1 Fv (3n 6 h) + 3 = 3n h 3, a contradiction to n 3, h 1. Case ) N V F 1 v Fv (F v 1 Fv ). Let g = N V F 1 v Fv (F v 1 Fv ), Let u Fv 1 Fv, without loss of generality, suppose u Fv 1 Fv. Since N V F 1 v Fv (F v 1 Fv ), suppose there exists a vertex v V Fv 1 Fv such that uv E(u) F e. Since the number of the faulty edges is not more than h, obviously, d G (v) n h. Since Fv 1, Fv are indistinguishable, at most one neighbouring vertex of v is in Fv Fv 1 and the rest neighbouring vertices of v in Q n F e are in Fv 1 Fv. So Fv 1 Fv n h. Let G = Q n F e. Let S = N G (v) (Fv Fv 1 ) and S = g, then g 1. In this case, we consider the following two subcases depending on the size of g. Subcase. 1 g = 0 Since Fv 1, Fv are indistinguishable, then N G (v) Fv 1, Fv 1 N G (v) n h and combining with Fv 1 n h, so Fv 1 = N G (v) = n h, F e E(v), Fv Fv 1 = N G (v) u. If N V F 1 v Fv (u) v, suppose w N V Fv 1 F v (u) v (see Fig. ). Since Fv 1, Fv are indistinguishable, w has at most one neighbor in Fv 1 Fv or Fv Fv 1 and no neighbor in u F 1 v w v F v G F 1 v F v F e Fig. 1: An illustration of the proof of N G F 1 v F v (u) v of Subcase.1. V F 1 v F v. Since F e N(v), N G (w) (F 1 v F v ) n. Therefore, combining with lemma 4.1, for n 4, we have n h F 1 v = F 1 v F v + F 1 v F v 1 + N F 1 v F v (v) N F 1 v F v (w) 1 + N F 1 v F v (v) + N F 1 v F v (w) 1 1 + n h 1 + n 1 a contradiction to n 4. If N V F 1 v F v (u) v =, since F v F 1 v = N G (v) u, u, v don t have common neighbours in F 1 v F v because of the minimum length of all cycles of Q n equals 4 for n 3, then all neighbours of u aren t in F 1 v F v, namely, N G (u) v F v F 1 v, so F v F 1 v N G (u) v = n 1. It is easy to obtain that for n 3, n h F v = F 1 v F v + F v F 1 v N G (v) u + N G (u) v (n h 1) + (n 1) A contradiction. Subcase. g = 1 Let x be a neighbouring vertex of v in F v F 1 v, then F 1 v F v N G (v) n h. Combining with F 1 v n h, F v n h, we can obtain that F 1 v F v and F v F 1 v We will distinguish between the following two cases. Subcase.. 1 F 1 v F v = or F v F 1 v = Without loss of generality, we assume that F 1 v F v =. Let u, y F 1 v F v, then F 1 v F 1 v F v + F 1 v F v n h + = n h, so Fv 1 = n h, Fv 1 Fv = n h and F e E(v). Observe that this situation is similar to above mentioned subcase.1, with similar arguments, a contradiction can be obtained. Subcase.. Fv 1 Fv = 1 and Fv Fv 1 = 1 Let {u} = Fv 1 Fv, {w} = Fv Fv 1. Hence, v is adjacent to u and w and uv, vw are fault-free edges. Since Fv 1 Fv n h, Fv 1 n h and Fv 1, Fv are indistinguishable, N G (v) w Fv 1, d G (v) 1 Fv 1 n h, therefore, F e E(v) h 1.

5 If there exists a vertex x B, such that x N G (u), then the other end-vertices of all the fault-free edges incident to x have at least n 3 in F 1 v F v because of F 1 v, F v are indistinguishable. By lemma 4.1, x, v have exactly two common neighbours for n 3 if they have any. Therefore, for n 5, F 1 v = F 1 v F v + F 1 v F v N F 1 v F v (v) N F 1 v F v (x) + 1 N F 1 v F v (v) + N F 1 v F v (x) + 1 (n h ) + (n 3) + 1 n h + 1. This is a contradiction, so there is no fault-free edges between u and B except uv. By lemma 4.1, u is not adjacent to w, so, N G (u) v (F 1 v F v ). Therefore, for n 4, n h F 1 v = F 1 v F v + F 1 v F v N F 1 v F v (u) N F 1 v F v (v) + 1 N F 1 v F v (u) + N F 1 v F v (v) + 1 (n 1 1) + (n h ) + 1 a contradiction. Thus t e h (Q n) n h, the Lemma holds. Combining Lemmas 4.4 and 4.5, we have the following theorem. Theorem 4.6: Let h, n be positive integers with n 5 and 1 h n 1, then under both the PMC model and the MM model t e h (Q n) = n h. V. CONCLUSIONS In this paper, we propose the h-edge tolerable diagnosability as a measure for the diagnosis capability analysis of multiprocessor systems with both link and node faults. Then we study the h-edge tolerable diagnosability t e h (Q n) of the hypercube Q n under both the PMC and the MM model. We prove that t e h (Q n) = n h for n 5 and 1 h n 1. The h-edge tolerant diagnosability of other interconnection networks are still to be determined in future. REFERENCES [1] G. M. F.P. Preparata and R. Chien, On the connection assignment problem of diagnosable systems, IEEE Trans. on Comp., vol. 16, p. 848C854, 1967. [] J. Maeng and M. Malek, A comparison connection assignment for selfdiagnosis of multiprocessor systems, in Symposium on Fault-tolerant Computing, 1981. [3] A. Sengupta and A. Dahbura, On self-diagnosable multiprocessor systems: Diagnosis by the comparison approach, Computers, IEEE Transactions on, vol. 41, no. 11, pp. 1386 1396, 199. [4] G. Chang, G. Chang, and G. 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